In this talk, we see that neither bound is tight. Our new hardness result, obtained by a different application of the concavity method used also in the previous work, states that a success probability better than 0.3672 is not possible. Using both theoretical and numerical approaches, we improve the lower bound to $0.3384$, that is, give a protocol leading to this success probability. To ease the design of new protocols, we prove an equivalent formulation of the cryptogenography problem as solitaire vector splitting game. Via an automated game tree search, we find good strategies for this game. We then translate the splits that occurred in this strategy into inequalities relating position values and use an LP solver to find an optimal solution for these inequalities. This gives slightly better game values, but more importantly, it gives a more compact representation of the protocol and a way to easily verify the claimed quality of the protocol.
Unfortunately, already the smallest protocol we found that beats the previous 1/3 success probability takes up 16 rounds of communication. The protocol leading to the bound of 0.3384 even in a compact representation consists of 18248 game states. These numbers suggest that the task of finding good protocols for the cryptogenography problem as well as understanding their structure is harder than what the simple problem formulation suggests.
This is joint work with Benjamin Doerr.